Probabilistic physics-constrained reconstruction of three-dimensional flow fields

Accurate reconstruction of three-dimensional flow fields is essential for resolving fine-scale flow structures and understanding unsteady flow physics in ship and ocean engineering applications. However, while direct numerical simulation provides high-fidelity solutions of the Navier–Stokes equations, its prohibitive computational cost limits its applicability to large-scale and high-Reynolds-number flows. Reduced-order and data-driven reconstruction approaches offer improved efficiency, yet often fail to preserve inter-slice coherence, fine-scale vortical structures, and physical consistency when extrapolating beyond the training regime. To address these challenges from a physical-consistency perspective, this study investigates a physics-constrained framework for three-dimensional flow-field reconstruction. The governing physical constraints are embedded into the reconstruction process through a probabilistic formulation derived from the vorticity–velocity representation of the Navier–Stokes equations, enabling improved enforcement of incompressibility and dynamical consistency. A spatiotemporal reconstruction architecture is constructed to capture multi-scale spatial features and temporal evolution characteristics of unsteady flows, while a physics-informed attention mechanism is employed to enhance the recovery of fine-scale flow structures in regions with strong spatial gradients. The proposed framework is assessed using turbulent flow data from the Johns Hopkins Turbulence Database, transient boundary layer flows, and experimental tomographic particle image velocimetry measurements. Quantitative comparisons with high-fidelity reference data demonstrate that the physics-constrained reconstruction significantly improves the preservation of fine-scale flow structures and extrapolation performance compared with conventional deterministic approaches, indicating its effectiveness for physically consistent three-dimensional flow-field reconstruction at reduced computational cost.